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Abstract:
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This paper introduces a new way to compact a continuous probability distribution into a set of representative points called support points. These points are obtained by minimizing the distance-based energy statistic, which was initially proposed by Szekely and Rizzo (2004) for goodness-of-fit testing, but also has a rich optimization structure. Theoretically, support points can be shown to be consistent for integration use, and enjoy an improved error convergence rate to Monte Carlo methods. In practice, simulation studies show that support points provide sizable improvements in integration over both Monte Carlo and state-of-the-art Quasi-Monte Carlo methods, for moderate point set sizes and dimensions. We propose two algorithms for computing support points, both of which exploit the underlying difference-of-convex formulation to efficiently generate high-quality point sets. One important application of support points is in optimally reducing Markov chain Monte Carlo (MCMC) samples for Bayesian computation, which we demonstrate using two real-world examples.
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